#
Hyperfuzzy Ideals in BCK/BCI-Algebras^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (I)
- $\left(\right(x\ast y)\ast (x\ast z\left)\right)\ast (z\ast y)=0,$
- (II)
- $(x\ast (x\ast y\left)\right)\ast y=0,$
- (III)
- $x\ast x=0,$
- (IV)
- $x\ast y=y\ast x=0\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}x=y$

**Definition**

**1**

**([6]).**

- fuzzy subalgebra of $(X,\ast ,0)$ with type 1 (briefly, 1-fuzzy subalgebra of $(X,\ast ,0)$) if$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu (x\ast y)\ge min\left\{\mu \right(x),\mu (y\left)\right\}\right),\end{array}$$
- fuzzy subalgebra of $(X,\ast ,0)$ with type 2 (briefly, 2-fuzzy subalgebra of $(X,\ast ,0)$) if$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu (x\ast y)\le min\left\{\mu \right(x),\mu (y\left)\right\}\right),\end{array}$$
- fuzzy subalgebra of $(X,\ast ,0)$ with type 3 (briefly, 3-fuzzy subalgebra of $(X,\ast ,0)$) if$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu (x\ast y)\ge max\left\{\mu \right(x),\mu (y\left)\right\}\right),\end{array}$$
- fuzzy subalgebra of $(X,\ast ,0)$ with type 4 (briefly, 4-fuzzy subalgebra of $(X,\ast ,0)$) if$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu (x\ast y)\le max\left\{\mu \right(x),\mu (y\left)\right\}\right).\end{array}$$

**Definition**

**2**

**([6]).**

## 3. Hyperfuzzy Ideals

**Definition**

**3.**

- fuzzy ideal of $(X,\ast ,0)$ with type 1 (briefly, 1-fuzzy ideal of $(X,\ast ,0)$) if$$\begin{array}{c}(\forall x\in X)\left(\mu \left(0\right)\ge \mu \left(x\right)\right),\hfill \end{array}$$$$\begin{array}{c}(\forall x,y\in X)\left(\mu \left(x\right)\ge min\left\{\mu \right(x\ast y),\mu (y\left)\right\}\right),\hfill \end{array}$$
- fuzzy ideal of $(X,\ast ,0)$ with type 2 (briefly, 2-fuzzy ideal of $(X,\ast ,0)$) if$$\begin{array}{c}(\forall x\in X)\left(\mu \left(0\right)\le \mu \left(x\right)\right),\hfill \end{array}$$$$\begin{array}{c}(\forall x,y\in X)\left(\mu \left(x\right)\le min\left\{\mu \right(x\ast y),\mu (y\left)\right\}\right),\hfill \end{array}$$
- fuzzy ideal of $(X,\ast ,0)$ with type 3 (briefly, 3-fuzzy ideal of $(X,\ast ,0)$) if it satisfies (7) and$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu \left(x\right)\ge max\left\{\mu \right(x\ast y),\mu (y\left)\right\}\right),\end{array}$$
- fuzzy ideal of $(X,\ast ,0)$ with type 4 (briefly, 4-fuzzy ideal of $(X,\ast ,0)$) if it satisfies (9) and$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\mu \left(x\right)\le max\left\{\mu \right(x\ast y),\mu (y\left)\right\}\right).\end{array}$$

**Definition**

**4.**

**Example**

**1.**

**Example**

**2.**

**Proposition**

**1.**

- (1)
- If $(X,\tilde{\mu})$ is a $(1,4)$-hyperfuzzy ideal of $(X,\ast ,0)$, then$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)\ge {\tilde{\mu}}_{inf}\left(y\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)\le {\tilde{\mu}}_{sup}\left(y\right)\right).\end{array}$$
- (2)
- If $(X,\tilde{\mu})$ is a $(1,1)$-hyperfuzzy ideal of $(X,\ast ,0)$, then$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)\ge {\tilde{\mu}}_{inf}\left(y\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)\ge {\tilde{\mu}}_{sup}\left(y\right)\right).\end{array}$$
- (3)
- If $(X,\tilde{\mu})$ is a $(4,1)$-hyperfuzzy ideal of $(X,\ast ,0)$, then$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)\le {\tilde{\mu}}_{inf}\left(y\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)\ge {\tilde{\mu}}_{sup}\left(y\right)\right).\end{array}$$
- (4)
- If $(X,\tilde{\mu})$ is a $(4,4)$-hyperfuzzy ideal of $(X,\ast ,0)$, then$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)\le {\tilde{\mu}}_{inf}\left(y\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)\le {\tilde{\mu}}_{sup}\left(y\right)\right).\end{array}$$

**Proof.**

**Proposition**

**2.**

- (1)
- If $(X,\tilde{\mu})$ is an $(i,j)$-hyperfuzzy ideal of $(X,\ast ,0)$ for $(i,j)\in \left\{\right(2,2),(2,3),(3,2),(3,3\left)\right\}$, then$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)={\tilde{\mu}}_{inf}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)={\tilde{\mu}}_{sup}\left(0\right)\right).\end{array}$$
- (2)
- If $(X,\tilde{\mu})$ is either a $(1,2)$-hyperfuzzy ideal or a $(1,3)$-hyperfuzzy ideal of $(X,\ast ,0)$, then the following assertion is valid.$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)\ge {\tilde{\mu}}_{inf}\left(y\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)={\tilde{\mu}}_{sup}\left(0\right)\right).\end{array}$$
- (3)
- If $(X,\tilde{\mu})$ is either a $(2,1)$-hyperfuzzy ideal or a $(3,1)$-hyperfuzzy ideal of $(X,\ast ,0)$, then the following assertion is valid.$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)={\tilde{\mu}}_{inf}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)\ge {\tilde{\mu}}_{sup}\left(y\right)\right).\end{array}$$
- (4)
- If $(X,\tilde{\mu})$ is either a $(2,4)$-hyperfuzzy ideal or a $(3,4)$-hyperfuzzy ideal of $(X,\ast ,0)$, then the following assertion is valid.$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)={\tilde{\mu}}_{inf}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)\le {\tilde{\mu}}_{sup}\left(y\right)\right).\end{array}$$
- (5)
- If $(X,\tilde{\mu})$ is either a $(4,2)$-hyperfuzzy ideal or a $(4,3)$-hyperfuzzy ideal of $(X,\ast ,0)$, then the following assertion is valid.$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)\le {\tilde{\mu}}_{inf}\left(y\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)={\tilde{\mu}}_{sup}\left(0\right)\right).\end{array}$$

**Proof.**

**Proposition**

**3.**

- (1)
- If $(X,\tilde{\mu})$ is a $(1,4)$-hyperfuzzy ideal of $(X,\ast ,0)$, then$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \left\{\begin{array}{c}{\tilde{\mu}}_{inf}\left(x\right)\ge min\{{\tilde{\mu}}_{inf}\left(y\right),{\tilde{\mu}}_{inf}\left(z\right)\}\hfill \\ {\tilde{\mu}}_{sup}\left(x\right)\le max\{{\tilde{\mu}}_{sup}\left(y\right),{\tilde{\mu}}_{sup}\left(z\right)\}\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$
- (2)
- If $(X,\tilde{\mu})$ is a $(1,1)$-hyperfuzzy ideal of $(X,\ast ,0)$, then$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \left\{\begin{array}{c}{\tilde{\mu}}_{inf}\left(x\right)\ge min\{{\tilde{\mu}}_{inf}\left(y\right),{\tilde{\mu}}_{inf}\left(z\right)\}\hfill \\ {\tilde{\mu}}_{sup}\left(x\right)\ge min\{{\tilde{\mu}}_{sup}\left(y\right),{\tilde{\mu}}_{sup}\left(z\right)\}\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$
- (3)
- If $(X,\tilde{\mu})$ is a $(4,1)$-hyperfuzzy ideal of $(X,\ast ,0)$, then$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \left\{\begin{array}{c}{\tilde{\mu}}_{inf}\left(x\right)\le max\{{\tilde{\mu}}_{inf}\left(y\right),{\tilde{\mu}}_{inf}\left(z\right)\}\hfill \\ {\tilde{\mu}}_{sup}\left(x\right)\ge min\{{\tilde{\mu}}_{sup}\left(y\right),{\tilde{\mu}}_{sup}\left(z\right)\}\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$
- (4)
- If $(X,\tilde{\mu})$ is a $(4,4)$-hyperfuzzy ideal of $(X,\ast ,0)$, then$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \left\{\begin{array}{c}{\tilde{\mu}}_{inf}\left(x\right)\le max\{{\tilde{\mu}}_{inf}\left(y\right),{\tilde{\mu}}_{inf}\left(z\right)\}\hfill \\ {\tilde{\mu}}_{sup}\left(x\right)\le max\{{\tilde{\mu}}_{sup}\left(y\right),{\tilde{\mu}}_{sup}\left(z\right)\}\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$

**Proof.**

**Proposition**

**4.**

- (1)
- If $(X,\tilde{\mu})$ is an $(i,j)$-hyperfuzzy ideal of $(X,\ast ,0)$ for $(i,j)\in \left\{\right(2,2),(2,3),(3,2),(3,3\left)\right\}$, then$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{inf}\left(x\right)={\tilde{\mu}}_{inf}\left(0\right),\phantom{\rule{3.33333pt}{0ex}}{\tilde{\mu}}_{sup}\left(x\right)={\tilde{\mu}}_{sup}\left(0\right)\right).\end{array}$$
- (2)
- If $(X,\tilde{\mu})$ is either a $(1,2)$-hyperfuzzy ideal or a $(1,3)$-hyperfuzzy ideal of $(X,\ast ,0)$, then the following assertion is valid.$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \left\{\begin{array}{c}{\tilde{\mu}}_{inf}\left(x\right)\ge min\{{\tilde{\mu}}_{inf}\left(y\right),{\tilde{\mu}}_{inf}\left(z\right)\}\hfill \\ {\tilde{\mu}}_{sup}\left(x\right)={\tilde{\mu}}_{sup}\left(0\right)\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$
- (3)
- If $(X,\tilde{\mu})$ is either a $(2,1)$-hyperfuzzy ideal or a $(3,1)$-hyperfuzzy ideal of $(X,\ast ,0)$, then the following assertion is valid.$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \left\{\begin{array}{c}{\tilde{\mu}}_{inf}\left(x\right)={\tilde{\mu}}_{inf}\left(0\right)\hfill \\ {\tilde{\mu}}_{sup}\left(x\right)\ge min\{{\tilde{\mu}}_{sup}\left(y\right),{\tilde{\mu}}_{sup}\left(z\right)\}\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$
- (4)
- If $(X,\tilde{\mu})$ is either a $(2,4)$-hyperfuzzy ideal or a $(3,4)$-hyperfuzzy ideal of $(X,\ast ,0)$, then the following assertion is valid.$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \left\{\begin{array}{c}{\tilde{\mu}}_{inf}\left(x\right)={\tilde{\mu}}_{inf}\left(0\right)\hfill \\ {\tilde{\mu}}_{sup}\left(x\right)\le max\{{\tilde{\mu}}_{sup}\left(y\right),{\tilde{\mu}}_{sup}\left(z\right)\}\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$
- (5)
- If $(X,\tilde{\mu})$ is either a $(4,2)$-hyperfuzzy ideal or a $(4,3)$-hyperfuzzy ideal of $(X,\ast ,0)$, then the following assertion is valid.$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\phantom{\rule{3.33333pt}{0ex}}\Rightarrow \left\{\begin{array}{c}{\tilde{\mu}}_{inf}\left(x\right)\le max\{{\tilde{\mu}}_{inf}\left(y\right),{\tilde{\mu}}_{inf}\left(z\right)\}\hfill \\ {\tilde{\mu}}_{sup}\left(x\right)={\tilde{\mu}}_{sup}\left(0\right)\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

- (1)
- $(X,\tilde{\mu})$ is not a $(2,3)$-hyperfuzzy ideal of $(X,\ast ,0)$,
- (2)
- The nonempty sets $U{({\tilde{\mu}}_{inf};\alpha )}^{c}$ and $L{({\tilde{\mu}}_{sup};\beta )}^{c}$ are ideals of $(X,\ast ,0)$ for all $\alpha ,\beta \in [0,1]$.

**Example**

**3.**

**Theorem**

**3.**

**Theorem**

**4.**

## 4. Relations between Hyperfuzzy Ideals and Hyperfuzzy Subalgebras

**Theorem**

**5.**

**Proof.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

- (1)
- If $(X,\tilde{\mu})$ is a $(1,1)$-hyperfuzzy subalgebra of $(X,\ast ,0)$ which satisfies the condition (23), then $(X,\tilde{\mu})$ is a $(1,1)$-hyperfuzzy ideal of $(X,\ast ,0)$.
- (2)
- If $(X,\tilde{\mu})$ is a $(4,1)$-hyperfuzzy subalgebra of $(X,\ast ,0)$ which satisfies the condition (24), then $(X,\tilde{\mu})$ is a $(4,1)$-hyperfuzzy ideal of $(X,\ast ,0)$.
- (3)
- If $(X,\tilde{\mu})$ is a $(4,4)$-hyperfuzzy subalgebra of $(X,\ast ,0)$ which satisfies the condition (25), then $(X,\tilde{\mu})$ is a $(4,4)$-hyperfuzzy ideal of $(X,\ast ,0)$.

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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∗ | 0 | 1 | 2 | 3 | 4 |
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0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 0 | 1 | 0 | 0 |

2 | 2 | 2 | 0 | 0 | 0 |

3 | 3 | 3 | 3 | 0 | 0 |

4 | 4 | 3 | 4 | 1 | 0 |

∗ | 0 | 1 | 2 | a | b |
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0 | 0 | 0 | 0 | a | a |

1 | 1 | 0 | 1 | b | a |

2 | 2 | 2 | 0 | a | a |

a | a | a | a | 0 | 0 |

b | b | a | b | 1 | 0 |

∗ | 0 | 1 | a | b | c |
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0 | 0 | 0 | a | b | c |

1 | 1 | 0 | a | b | c |

a | a | a | 0 | c | b |

b | b | b | c | 0 | a |

c | c | c | b | a | 0 |

∗ | 0 | a | b | c |
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0 | 0 | 0 | 0 | 0 |

a | a | 0 | 0 | a |

b | b | a | 0 | b |

c | c | c | c | 0 |

∗ | 0 | a | b | c |
---|---|---|---|---|

0 | 0 | 0 | 0 | 0 |

a | a | 0 | 0 | 0 |

b | b | a | 0 | a |

c | c | c | c | 0 |

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## Share and Cite

**MDPI and ACS Style**

Song, S.-Z.; Kim, S.J.; Jun, Y.B. Hyperfuzzy Ideals in *BCK*/*BCI*-Algebras. *Mathematics* **2017**, *5*, 81.
https://doi.org/10.3390/math5040081

**AMA Style**

Song S-Z, Kim SJ, Jun YB. Hyperfuzzy Ideals in *BCK*/*BCI*-Algebras. *Mathematics*. 2017; 5(4):81.
https://doi.org/10.3390/math5040081

**Chicago/Turabian Style**

Song, Seok-Zun, Seon Jeong Kim, and Young Bae Jun. 2017. "Hyperfuzzy Ideals in *BCK*/*BCI*-Algebras" *Mathematics* 5, no. 4: 81.
https://doi.org/10.3390/math5040081